00:01
We're being asked to expand the given binomial using the binomial theorem.
00:04
Well, when we do this, our first term when we do the expansion will always just be the first term in the binomial, which in this case is q, raised to that exponent, which in this case is 9.
00:16
Now, moving forward, we're first going to have to deal with the coefficient, which in this case, to get the coefficient, what we do is we take our exponent, which is 9, and for the first one, we're going to divide it by 1 factorial.
00:29
Now, for our variables, for the first term, we're going to raise it to the 9 minus 1 power, and then for our second term in the binomial r, we're going to raise it just to the first power.
00:40
Now let's find the third term.
00:42
Well, to find the coefficient, remember how we started with the 9 before, so we're going to have 9, but now we have to multiply it by 1 less than 9, which is 8.
00:51
And then we're going to divide it by 2 factorial.
00:54
So notice our factorials each time go up by 1.
00:57
Now, for our variables, we're going to have q.
01:00
To the 9 minus 2 power, and then r to the second power.
01:05
And we're going to keep this pattern going until we finally get to that r to the ninth term, as that will be the last term.
01:11
So for our next term, for the coefficient, we're going to have 9 times 8, times 7, all over 3 factorial.
01:20
Well, then we're going to have q to the 9 minus 3 power, and r to the 3 power.
01:27
Now, for the next term, we're going to have plus 9 times 8 times 6.
01:32
Seven, oops, put that in parentheses, times six, all over four, factorial, and then q to the nine minus four, r to the fourth.
01:45
Now, we're going to go on to the next line.
01:48
So we're going to have plus, and for our coefficient, we're now going to have nine times eight, times seven, times six, times five, all over five factorial.
02:00
And then we're going to have q get raised to the nine minus fifth power, and then r will get raised to the fifth power.
02:07
Now, for the next term, our coefficient will be 9 times 8, times 7, times 6, times 5, times 4, all over 6 factorial, and then we're going to have q raised to the 9 minus 6 power, r to the 6 power.
02:25
Now, for our next coefficient, we're going to have 9 times 8, times 7, times 6, times 5, times 4, times 3, all over 7 factorial.
02:38
And then for our variables, we'll have q to the 9 minus 7, and then r to the 7th...